Quadratic forms in design theory

Tilburg University

Quadratic forms in design theory

Coster, M.J.

Publication date:

1994

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Coster, M. J. (1994). Quadratic forms in design theory. (Research Memorandum FEW). Faculteit der

Economische Wetenschappen.

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1994

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QUADRATIC FORMS IN DESIGN THEORY

M.J. Coster

Research Memorandum FEW 635

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Quadratic forms in Design Theory.

M.J. Coster

Janu~r,y 12, 1994

Abstract

This report describes the Grothendieck (ároup for rational congruence classes of positive definite integral matrices. Thc main result is an explicite diagonal matrix for each class of this Grothendicck Group. We give some applications in design theory.

Keywords: quadratic forms, designs, positive definite integral matrices,

decompo-sition.

1

Introduction.

Let M be an integral positive cíefinite symrnetric matrix. In this paper we consider conditions for which M can be writt.en as M- AAT, where A is an integral square matrix ( in this case we say that M is decorreposable). There is a one-one relation between integral positive definite symrne.tric rnatrices and positive definite quadratic forms with integral coefficient,s. Though the theory can completely be described in terms of quadratic forms, we prefer the use of matrices. 'I'he reason to do this is the fac.t that our applications deal with matrices. The theory on quadratic forms can be found in [1, 3]. In these books

the theory is described in its generality. We consider just a special case.

We will denote by S the set of positive definite integral symmetric matrices. We denote by C the related Grothendieck group. 'I'his group will be defined formally in Section 3. In Section 4 we consider the structure of Cj. This structure is completely described by Theorem 4.3. As a consequence of this theorem we can write each element g of ~ uniquely as a surn of infinitely many generators. We will show as a consequence

that G- Cz ~ V~ ~ C~, where V9 is the 4- group of Klein.

'1'he second part of the paper deals wit,h some decomposition problems which arise in

design t,hcory. Usually one solvcs thcse problems with the Hasse--Minkowski invariant,s.

We will show that these problerns can bc solved easily without using the original Hasse Principle. In this paper we will mainly develop tools for design theory. We end with a general application. In Sect,ion 6 we will apply our method to the lattice graph. An application Lo C~uasi Symmetric designs is given in [4].

2

Notation.

'1'he following notation will be used freyuently in this paper. Most notations will be explainecí again in t.he text.

~-~ The Legendre symbol.

n` The squarefree part, of an integer n.

S is the sct of positive definite syrnrnetric intcgral matrices.

CJ is the Grot,hendieck Group associated to S. "Q3" is a rnatrix addition.

"l)" is a rnat,rix addit,ion. ~~,.,,,_{- is t, ic cougrucncc rclation.}

(A~ is t,he class of the matrix A. (if A is a positive integer then (A) -((A)~.

(~-p~ is the class of the matrix (~pq) EIj (~q) Efl 3(~), where q is a prime depending on p.

1„ is the n x n-identity matrix.

J„ is the n x n all one matrix. j is the all 1-vector.

It ri - ~n ~ Jn. lV~n _{- ( n ~ í ),n - Jn .}

3

Basic Theory.

Let S be the set of positive definite symmetric integral matrices, including the empty set clement. We define an addition on S in the following way. Let A, B E S then we define

A~ B-~ Ó ~ J.([f A is an m x m-rnatrix and B is an n x n-matrix then A~ B is

an (m f n`) x(m -~ n)-matrix, with on the diagonal the original matrices A and B). The sct S with the addition (p is a semigroup, (i.e. it satisfies the grouplaws, except of the existence of an inverse), with unity 0, the ernpty set element. Now we define a relation. i.et A and B be two elements of S of dirnensions m and n respectively. We say A- B if there exists an integral k x k-matrix CZ such that Q(A ~ Ik-,,,)QT - B~ I~-n. The relation - is an equivalence relation, called rntional equivalence relation (see [1, 3]). The result is based on the Wif,t cancelat,ion. We~ denote the class of the rnatrix A by (A). It is well-known that each equivalence class contains a diagonal matrix (i.e. a matrix with zeroes outside t,he diagonal), see [1]. We denote the 1 x 1-matrix equivalence class ((a)) sirnply by (a). Each class (A) can be writ,ten as (A) - t~t(a;) for some positive integers a;. Notic.e t,hat (In) -(1) -(0) - 0. The following Lemma can also be found in [1].

Lemma 3.1 (,et a, b E 71~0. Th.err. we Irreve

Í~) (a) ~,f) (b) -(a f b) Ef) (ab(a f b)).

Ilence the seL S wit,h t.hc opcra.tion ~;li and equivalence uuder - is a sernigroup. Wc denotc~ t.Lis se~nrigroup by ~(hr~nce C~ -(S~ -, ~i))). 'I'his is the Crolfterzdieck yroup. '1'he following Icninra, shows tha.l, ~ is indccd a. group.

Corollary 3.`l Lcl a, b, c~, n E rl,~i~. 'I'Itcn roc leavc

(1) 2(c(ct~ f h~)) - 2(c),

(2) 4(n) - 0.

Proof.

(1) 2(c) - (a2c) ~ ( b2c) - (c(a2 } G2))~~ (a2b2c(a2 -f bz)),

(2) 0 2(a2 f 62) m 2(cz ~ dl)

2(a2~h'~~c2fd2)CD2((a~fb2)(c2~d2)(azfb2~c2~d2))

-4(a2 f 62 ~ c2 f d2). Now apply Legendre's Theorem which says that every positive

rational integer can be written as sum of four squares.

Corollary 3.2 says that the inverse of (a) is 3(a). Hence each element of ~ has an inverse.

Therefore ~ is a group. 'I'herefore we are able to define e( a) - 3(a). We extend this

de(inition t,o El(a) - 3(a) -(- a). 'I'his de(inition becomes usefull because of t,he following generalization of Lernma 3.1. '1'his lernrna is new as far as we know. '1'he advantage of using this lemma is that it deduct,s a lot of c.alculations in the remainder of the paper. Lemma 3.3 l,et a, b E 7I`{0}. Then we have

(1) (ab2) - (a),

(2) If a f b ~ 0 then (a) ~ (b) - (a -h b) ~ (ab(a ~- b)).

Proof. _{The extension of hetnrna 3.1 ( 1) is easy to prove and the proof is left to the} reader. We will proof Lemrna 3.1 (2). We have to distinguish 4 cases namely ( i ): a 1 0 andb)0; ( ii): aGOandbGO; ( iii): afh)OandabGO; (iv): afbGOand

ab G 0. Case ( i ) was provcd in Lernrna 3.1. ['or case (ii ) multiply case (i ) by -1. In

casc~ ( iii ) wc~ niay assnnic~ 1,haL a 1 0 ancl b G 0. Subst,it,nt,c 6 by -c and notice that,

(a - c) CD (c) -(a) Q) (ac(a - c)). l~or proving casc ( iv ) mult,iply casc ( iii ) by -1. ~

4

The Main Theorem.

'1'he main theorem describes the Grot,hendieck Group in our special case (positive definite). We introduce the p-excess (cf. [3]; It is possible t,o prove the Main Theorem avoiding the p-excess, but thc p-excesses make t,hc proofs shorter.) Let g E~, g- t~k(ak), with ak integers. E~'or p an odd prirnc we define thc p-excess by

0 rnod 8 if p Xa', i c xcc 55 a 1~ - 1 rnod 8 if a p p - 1,

7' ~ "~~'(( ))

-p~ 3 mod 8 if a pp --1.

4

Now p-c~xcess(g) -~?r-exeess((a,~)). 'l'he '?-c~xeess is defined by

-~ l- a mod 8 if a odd 2-c~xcess((a) )

1(1 - z)~ rnod 8 if aevcn. (2) Now 2-exccss(g) - ~ 2-exccss( (ak) ).

Note. It is easy to verify by straight calculation that p-excess((a) ~(b)) - p-excess((a f

h) ({) (ab(a ~ 6))) rnod 8, for each prime p.

In [3], Conway and Sloane prove a theorem about p-excesses. Here we will give the positive definite versiorr.

Theorem 4.1 Get g, h E~. STLppose g - r~(a,) and h-[~(b;). Then g- h is equivalent lo ~ a; .~ b; is a square and p-excess(g) - p-excess(h) mod 8 for all primes

p~

Note. In I,hc gencral casc il, is irnportant to considcr the (-1)-exccss. Ilowcver in our sil,uation wc~ havc (- I)-cxcess - 0.

Let p and q be t,wo primes then wc like to express (pq) in tcrms of (p) and (q). Por example (55) - (5) ~ (11) and (`Z1) - (7) (~ (3). I3ut (10) cannot be expressed in terrns of (2) and (5) as dcscribed above. On t,hc ol,hcr hand notice that (10) [~ (2) -(15) ~(3). Our goal of the Main 'I'heorern is to express eacFr class of t,he Grothendieck Croup CJ in terms of (p), where p is a prirne. In order to split (10) in terms of (2) and (5), we introduce the

syrnbol (p~.

Definition. Let p- 1 mod 4 be a prime. ~I'hen (p~ - (pq) ~(q), where q~ 1 mod 4 is an arbitrary prime for which (p) --1. Notice that such a prime q always exists, cf. [6],Thm. 15 and Thm. 84.

Note. The symbol (p~ is well defined, independent of the choice of q. To see this, suppose

(p~ (pq) ~ (q). `I'hen qexcess((p~) qexcess((pq)) ~ qexc,ess((q)) 2q 2 f 4

-0 mod 8.

We extend the definit,ion of (p~ t,o

Definition. Let p- 1 mocí 4 ancí ~ a non-zero integer then we denote by

(~ . p~ -(,~pq) (~ (~q) (~ ( a), wherc q is a prime as was defined in the previous definition.

It is casy to derive t,he following laws of addit,ion.

Lemma 4.2 Lr.t p and q be lroo priraes and let ~ 6r. a non-zero inleger. 'lhen we havc

(z ) (~`1rq)

-(~P) ~ (~9) H (~) ~f ~ - ~q~ - 1

(~p)~~(~9)(}~(a) if p -1,andp-4-3rnod4

(,~.pl~~)(~.qlir(`) if p --landp-4-1mod4

5

( ~ - p~ ~.C3 (~y) ~ ~ (.~) if

(~q ' 7~~ - (~p) H(~q) ~F~ (~) i[ p--1 and q - 3 mod 4

(~P)~~~(~'q~~-~(~) ~l ~ -- landq-l mod~

Proof. ~111 c~yual.ions can bc vcri(icd Iiy p-cxccsscs or by straight calculation. We will show by- straight calculation that if ~n) -- 1 ancl ~ - q- 1 mod 4 t,hen ( .~~q) -(~ . pl m(~ .

q~ (~ (.~). We need a lemma which will be proven in the following section. First rtotice that

there exists a prirne r such that, r - 3 rnod ~, (p) --1 and (9) --1 (See [6],Thm. 15 and Thrn. 84). Lemrna 5.1 tells us that (~yr) ( ,1qr) m(~yq) ~(~). Hence ( .~pq)

-(~yr)(~(~qr)~(~) - ((~Pr)~(~r)~~(~))iï)((~9r)~(~r)e(~))~2(.~qr)ff32(~r)E13`l(.~)~(~).

Now apply Lemma 3.2, (2). '1'hen wc havc -(~pq) -C ~. p~ m(~ . q~ ~ 4(~r) E3 (,~). 0 1?ach clen~ent of Lhe Crothendieck Group C~ can be represented in cíiagonal forrn. '1'he "1'heorern bclow shows Lhat, there is a uniquc reprcaentation in terms prime elernents. Theorem 4.3 ( Main Theorem.) l,ct ~S hc lhe sel of positive defiitile integral syrrtrnetric

rnatrices. _{Get C-(S~ ?', (fj) be the associated Crothendieck Group. l,et g E~. Then g} can be written in a unique way as follows.

.9 - ó (2) m ~(h[ (P~) ~ Eti (P~~) ~ ~ rIi (4i), (3)

wftere ~; and q~ a.rr, prirnes witit p; - 1 rnod h aud q~ - 3 mod 4, where b, S; and e, are 0 or 1, while rt~ E{0, 1, 2, 3} .

Proof. Let g-~(a~). I1, is suliicienL to prove that (ay) can be written in the forrn of hormula ~1.3. Suppose a~ -~~;. `I'hen il, is a consequence of Lemma 4.2 that ay can be writt,en in t.he forrn of h'orrnula 4.3.

The uniqucness of F'orrnula 4.3 follows immediately frorn '1'heorem 4.] or can be proven on induction.

O

Note. The Grothendieck Group can be seen as CZ~VA"o~C~, where C,~ - 71,~n'17 and Vq is the ~l-group of Klein. CZ is related to the prime 2, V4 is relat,ed to primes p- 1 mod 4 and C~ is related to primes q - 3 mod 4.

We conclude wit,h 3 usefull corollaries which follow immediately frorn the Main Theo-rcrn.

Corollary 4.4 Get g E~. Suppose q-~k(ak), with ~ ak is a square. 'lhen

written uniquel,y as

.9-~fi~((P~)~(Pá~) ~) 2~e~(9i)~

.9 can be.

(i

Corollary 4.5 I,cl g E~. .Suppasc ,q - (~l) tii 2~A(ak}. If g - 0 tlcerc det ~l must be a

,tiquare.

Corollary 4.6 Lcl g E G. Suppose g - 2~k(ak}. Thr.n 9 - Z ~(9~},

2vhere q~ are prirnes wilh g~ - 3 rnod 4.

5

The relation with Diophantine Equations.

'1'he following lemmas deal with the rclation bctween eyualities in the Grothendieck Group and related lliophantine equations.

Lemma 5.1 l,ct a, b ctnd c Gr. posilive inlr.gers whicit are squarefree and relalively prime.

in pairs. Lcl .~ be an arhilrary nort zcro inleqer. Then the following lhree slatements are equivalcnl:

(1) Iór all prirnes p dividing a lhe Leyendre symbol ~p`~ - l, for all prirnes q dividing b, ( 9)- i and for all primes r dividing c, (-Tb~ - 1.

(2) (aac} ~ (abc} - (aab) ~ (a),

(3) aX2 f bY2 - cZ2 has a non trivial inte,qral solution in X, Y and Z,

Lemma 5.2 l,et a, b and c be pnsitive integers which are squarefree and relatively prime

irt pairs. I,e.t .~ be an arbitrary vton-zero inle,qer. Then the following three statements are equivalent:

(1) For all printes p dividing a, ~-~`~ - l, f~or all primes q dividing 6, ~-q`~ - 1, For all printcs r dividi.rtq c, ~-re~ - 1

(2) (~ab) m (,1ac) ~ (.~bc) ~ ( .~) - 0.

(3) aX2 f bY2-~c72 - ahcW2 has an integral solution in X, Y", Z and W with XY'L ~ 0,

Proof. (Lcmrna 5.1 and Lernrna 5.2.)

(1)t-~(2) Can be derivcd frotn thc p-excesscs which were dcfined in t,he previous section, (See also [3], p~. 372, Thcorem 4.) First prove (1)t~(2) in case that ~- 1. Then apply the definition of F~ in order t,o show t,hat ~- 1 can be replaced by an arbitrary non zcro integcr.

(1)t~(3) This cquivalcnce is bascd on a t,hcorem of Legendre. See [5], pp. 423-433, [7]

7

Lemma 5.3 l.~cl a, b and c be integers whicie are squarefree and retativety pri~rze in pairs.

And s7cppnse that (ac) m(br) -(ab). Wc have (1) If abc ~is~ o~ld fhr~n. a. - c tnud 1 or b - c tnod -1, (2) lf c is cvcn tleen a ~ b - 0 niod 23 or a f b - c rnod 8,

(3) _{If ah is even, say a is even then a f b- c mod 8 or b- c mod 8.}

Proof. We calculate the `l-exeess of (ac), (hc) and ( ab) respectivily. We distinguish the same cases as in thc lemma.

(1) _{The2-excess identityreads ( l-ac)-}(1-bc)-(1-ab) - 0 mod 8. Sincec~ - 1 mod 8,} wehavec2-ac-bc}ab-(c-a)(c-h)-0mod8. Noticethata-candb-c are ~~ven. Ilence a- c rnod ~ or b - c mod 1.

(2) _{Let c '~t'. Now ' I'hc, 2cxass idcntity rcads t(1 aC)2 f z( I bC)Z (1 ab)} 0 niod 8 or a~('~~ ~ ti~C'~ 2aC `L6C ~ 2aG 0 mod 16. Noticc that a~C~ ~ b~C~

-a~ -t ti~ mod ifi. llcna, (a f Ir -(.')~ - (,"~ ~nod lfi.

(3) _{Let a 2A. Now The 2excess identity reads i(1 Ac)~ 1 (1 be) z(1 Ab)~} 0 mod 8 or A2cl Alf,l 2Ar. 2bc ~ 2Ab f 2 0 mod 16. Notice that A2c2 ~ AZb~ -b2 -}- c~ - 2 mod 16. Ilence (A ~ h - c)2 - AZ rnod 16. ~

Another proof can be, given by applying Lernrna 5.1, (2)t~(3). Notice that XZ -0, 1 or 4 mod 8.

Lemma 5.4 L,et a, b and c be int.egers which are squarefree and relatively prinze in pairs.

And suppose that (ab) ~(ac) ~(bc) - 0. Then (1) If abc is odd then a - b- c mod 4,

(2) If abc is rvcn, srcy a, i.s cvcn, then G-} c- 1 rnod 8 or a~- b-F c- 4 mod 8.

Proof. 'I'he. proof is comparable to the proof of the previous lernrna.

6

Some applications.

In this Section we will give two applications. The first application is t,he well-known

'I'heorem of liruck-Chowla-Ilyser ( see [2]. The second application deals with the Lattice graph. For another application see [4].

Before we give the two applications we will prove a lemma which expresses (aIn -{-~iJn) in terms of diagonal elements.

Lemma 6.1 l,et Iín - In ~ Jn and Mn - ( n -{- 1)In - Jn, then we have

(1) (ai,n) -(n f 1)(a) ~~ (a(n f i)),

(2) (o~„ ~ ,íiJ„) - n(~) E~ (an) ~ (n(~in f n),

8

Proof. Let, Q„ - ~ 1 T~ i~ bc an n x n-rnatrix. We easily calculate that Qn(al„ -F

J

nlí„-, 0 ~lín 0

~~J,~)Q„ - ~ _{OT n(~iri.~-cr) ~.} I?spc~cially M,~fi~InfrM.~ti - ~ _{Or ~(n~- 1) ~~}

IIence (n, -~ 1)(.1) - (.~L~„) (~ (.~(n, f I)), which proves (1).

'I'hc proofs of (2) and ( 3) can be found from (cYI„ f~iJn) -(cYlí„-r) m(n(Qn ~ a)). 0

We givc anothcr proof oL

Theorem 6.2 ( Bruck, Chowla and Ryser) I,el D be a symmetric 2-(v, k, ~)-design.

Then v, k and ~ satisfy the folloiuing identity: If v is even then k-~ is a sqnrere. If v is odd then

v-1

(k - ~)Xz ~- (-1) z vYz - 7z (5)

has a non-lrivial integral solution in X, Y and 'I,.

Proof. Let A thc associated incicíence matrix. Then AAT -(k -.~)I„ }~J,,. Therefore

((k -~)I„ ~- ~Jt,) - 0. Now apply Lemma 6.4. We conclude that

z~(k - a) ~3 (v(k - a)) c~ (v(va } (k - ~))) - a

(s)

Notice t,hat v~ -~ (k -~) - kz. Now Formula 6 can be read as

v(k - a) ~ (v(k - ~)) ~ (v) - 0. (7)

If v is even t,hen the number of diagonal elements in which the factor (k -~) appears

v-7

iti odcl. Ilence k-.~ rnust bc~ a squarc~. If v is odd Lhen v-(-1) ~ mod 4. Hence l~ornnrla (i can bc writ,tcn in t~hc~ forrn

„-r

(v) ~ (-1) ~ (k - a) - (v(k - ~)). (~)

Now apply Lcrnrna ~i.l. _D

of Gn corresponds t,o the edge l'~1;,~~, of k,,,,~. We denote by Ln the adjacenc.y matrix corresponding to G,,. 'I'he cigenvalucs of L,~ are

~~(Tt - })~~ ,~7A - ~~~tn~rl and ~-~~tn-~l~ .

In this section we will give clecornposability condit,ions for a matrix of the form

II.,~ - cr I ~ í~ L„ f-y J.

A main role is played by t,he cigenvalues of II.n. It, can be verified easily that the eigenvalucs of II.n are

r'o - ~-~Z~3(n-1)}yn2 r~ - n~~3(n-2) rz - cY - 2~i

We will consider I[,n being a function of ro, rr, r2. Our main theorem decompose

~I,n(ro, rl , r2).

Theorem 6.3 hel ILn - II,,~(r~,r~,rz) hc defiracd as above. 7'hen we have

n-i

(1[.n) -(ro) m Zn(rr ra) (~ 2(rr )({~ 2n(r2) ~ 2~ i(r2i).

i-5

As a consequence of this theorem we have the following corollary:

(q)

Corollary 6.4 Let 1[.n - 1Ln(ro, rr, r2) be defined as above. And s~appose that II~n is decomposable. Then ro must be a sq~care and 1Ue have

(i ) If n - 0 mod 4 lhen 2(rr) E13 P(n) - 0, (ii ) If n- } mod 4 then 2(n) m 2(r2) ~ P(n) - 0, (iii ) If n - 2 mod 4 then 2(rl) ~ 2(r2) F~ P(n) - 0, (iv ) If n - 3 mod 4 lhen 2(n) Ef) P(n) - 0,

n

2

where P(n) - 2~(2i - 1).

i-a

Proof. Sincc ll,n is decomposable, Formula 9 rnust be equal to zero. I~xcept of the first term of the right-hand part (ro) all terms appear in pairs. Therefore ro must be a square, (see Corollary h.5). We consider four cases depending on n mod 4. We share the terms

2n(r~n) and 2(rr) which results, applying Corollary 4.6, in 2(rl) for even n and 2(n) for

odd n. '['he other terms can be sirnplified to P(n) or 2(r2) ~ P(n) (depending on n). ~

Note. For n 1 8 the surn P(n) will nevr.r be equal to 0. 1~(n) grows rather fast. For cxamplc P(8) - 2(7), P(12) - 2(7} (B 2(11) and P(20) - 2(3) ~ 2(7) (D 2(11) (D 2(}9).

10

Lemma 6.5

(II~n(r0, 1~1,1~2)! - (10~ l~ Z~1'I (1LL,i-r - .~)~ (f) (r2(~3L~i-1)z ~- L~n-1 ~ J)).

Proof. We are searching a mat,rix C~ such that C~T ILnC~ is of the form as desired in the lemma. We build up (,~ frorn three subrnatric.es fó, h'i and h2i such that Q -( L~o~Ii~F'2). We construct, I'; frorn E; which are de(ined by

~'o !', E2 'J n2 (2nl f nL,n - 2J) nz ((n~ 2n) I nT n~.L) -II~n(í, 0, 0), II.n(0, 1, 0), II~n(0,0, 1).

'f'hcsc mat,riccs L;; satisfy F,; l;~ - b;3 h;;. Lct l be t,he all one vec.tor, let (I be an n~ x 2(n-1) niairix clc,(inc~d liy (f - (r~;i), whc~rc~

1 if L~ E L3; for 1 G j G 2(n - 1) and

u;~ - 0 Ci Cnz- 1,

0 clsc,

and let V be an n2 x ( rc - 1)1 rnatrix de(ined by V-( v;~), where

v;~ - ~ - í .

Now we construct the matrices I; by

L'o - nLol,

1', - F, U,

I:'2 - F,z V.

Since II.nF,; - r;E; for 0 C i G 2 we get

C~T II,n(ro, rr, ra)C~ - rol~ó j'o ~ r, l~,T Fr 03 r2F2 I~z.

1 if cithcr i- 1

or i- an ~(b ~ l) and j-(a - 1)(rc - 1) ~- b,

-1 if either i- b~ l and j-(a - 1)(n - 1) ~- b,

II

`I'his proves the lemrna.

Proof of Theorem 6.2. Wc, will first provc~ a special case, narnely (~ll,~((n~ 1)l, rz~-1, 1)). We use induclion. l~or n. -`l wc~ get. (aQ.~(S1,3, I)) - 2(a) (D 2(3~). Now we apply the previous letnma. We have

(Q,,,(~(rz f l)z, a(n f I), ~)) -(a) tD 2(~(n1„-r -.I)) Ef3 (~(311n-ih f L„-i f .l )).

We apply Lcmrna (i.d wliicli says that ( ~(nl,~-~ - J)) - n(.1n) ~(~). 13y induction on n we get

(~.n(a(n ~ 1)~, ~(r~ ~ 1), a)) - ~(~) ~~ 2(3~) ~ ~ ~;3 (i(~i) ~ (~))

- 2(n - 1)(.~) ~ 2~~s i(~i).

Now we can prove the t}ieorem in its gencrality. We get

(Q~n(ro, ri, rz)) - _{(ro) ~~ Z(ri ('~zln-~ - .I )i Ep (r~(3I(n-r)~ ~ Ln-r f ~1))}

-(ro) Cf) 2rz(ri n) íD 2(ri) (}~ 2n(rz} (I) 2 t~n-5 a(r2t)- p

'I'his Icads t,o nccessary condil,ions for squarc part,ial balanced designs sirnilar to the ones of S.S. Shrikhandc ancí N.C. Jain (cf. [9]).

References

[1] J.W.S. (~assels Ilational G~uadratic Íorms, Ac,ademic Press, London, ( 1978). ['?] S. ('Iwwla. c~ii ll..l. I~ysc~r C.`nrzzbitzalorio.l Nrnblems, Can. .1. Math. 2( 1950), 93-99.

[3] J.H. Conway and N..I.A. Sloane .S~Izere Packings, Lattices and Groups, Springer-Verlag New York, (1988).

[4] M.J. Coster and W.I1. Hacmers (~uasi-symmetric Designs related to the Triangular

graph, Research rnemorandurn FI?,W 596, Tilburg Univ. 1993.

[5] L.I~. Dickson Ilislory of thc lheor,y of Numbers, Volume II, Chelsea Publishing Corrr-pany, New York, 1966.

[6] G.H. IIardy and I~.M. Wright An Introduction ta the Theory of Numbers. Oxford University Yress, Oxford, (1983), fifth edition.

[7] L.J.Mordell, On the equation a.x~ } b,y~ - cz~ - 0., Monatshefte fur Math. 55 (1951), 323-327.

[8] L..l.Mordcll, Uiophanlirzc l;'qualinns Acadcrnic Press, inc., London, New York, l9(i9.

[9] S.S. Shrikhande and N.C. .lain Thc Norz e,rist,ence of some partially balanced

inr.om-pletr bloc~k rlesiyrz.. rmith Lahin. .~qaarr lype association schcrne Sanlchga A'l1 (1962),

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iv

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